Question

In Exercises \(17-20\), a closed curve \(C\) that is the boundary of a surface \(S\) is given along with a vector field \(\vec{F}\). Find the circulation of \(\vec{F}\) around \(C\) either through direct computation or through Stokes' Theorem. \(C\) is the curve whose \(x\) - and \(y\) -values are given by \(\vec{r}(t)=\) \(\langle\cos t, 3 \sin t\rangle\) and the \(z\) -values are determined by the function \(z=5-2 x-y ; \vec{F}=\left\langle-\frac{1}{3} y, 3 x, \frac{2}{3} y-3 x\right\rangle\)

Short answer

The circulation of \( \vec{F} \) around \( C \) is zero.

Step-by-step solution

Understand the Problem Statement

We have a closed curve \( C \) that is the boundary of surface \( S \), and we need to find the circulation of the vector field \( \vec{F} \) around \( C \). The problem can be solved either by direct computation or using Stokes' Theorem. The vector field given is \( \vec{F} = \left\langle -\frac{1}{3} y, 3x, \frac{2}{3} y - 3x \right\rangle \).

Parametrize the Curve C

The curve \( C \) is parametrized by the vector function \( \vec{r}(t) = \langle \cos t, 3 \sin t \rangle \), where the \( z \)-values are given by the surface function \( z = 5 - 2x - y \). Plug in the parametric equations: \( x = \cos t \), \( y = 3 \sin t \), then \( z = 5 - 2(\cos t) - 3(3 \sin t) = 5 - 2\cos t - 9\sin t \).

Compute the Line Integral over C Directly

The line integral for circulation is \( \oint_C \vec{F} \cdot d\vec{r} \). Calculate \( d\vec{r} = \left\langle -\sin t, 3\cos t, \frac{d}{dt} (5 - 2\cos t - 9\sin t) \right\rangle\). The last component simplifies to \( 2\sin t - 9\cos t \) after derivation. Substituting \( \vec{r}(t) \) into \( \vec{F} \), perform dot product and integrate over the interval \( t \) from 0 to \( 2\pi \).

Use Stokes’ Theorem

Alternatively, Stokes' Theorem states: \( \oint_C \vec{F} \cdot d\vec{r} = \int_S (abla \times \vec{F}) \cdot \vec{n} \, dS \), where \( \vec{n} \) is the normal to surface \( S \) and \( abla \times \vec{F} = \left\langle 0, 0, 0 \right\rangle \), since the divergence is zero, the circulation around \( C \) is zero.

Key concepts

Vector Fields

A vector field is essentially a function that connects to each point in space a vector. Imagine assigning a tiny arrow to every point on a map to reflect speed and direction, like how wind flows over a surface.
This is exactly what a vector field does, but in mathematical terms and often in three dimensions. In our exercise, consider the vector field \( \vec{F} = \langle -\frac{1}{3} y, 3x, \frac{2}{3} y - 3x \rangle \).
Here, \(x\) and \(y\) are variables, and for any given point, you substitute in these values to find a specific vector. Understanding how vector fields work is crucial for solving equations involving them, especially in relation to concepts like line integrals and integrals over surfaces.

Line Integrals

Line integrals extend the concept of integration to functions along a curve in a vector field. Think of it as measuring how much a vector field contributes to movement along a specific path.
The line integral of a vector field \( \vec{F} \) along a curve \( C \) is written as \( \oint_C \vec{F} \cdot d\vec{r} \). This computation evaluates the sum of the vector field values at every point along the curve, multiplied by infinitesimally small segments of the curve. In our exercise, the idea is to determine this integral either directly or using Stokes' Theorem.Direct evaluation involves calculating \( d\vec{r} \), the differential of the curve, and then finding the dot product with \( \vec{F} \). This is summed over the curve's path, giving insight into the vector field’s effect along the curve.

Parametrization

Parametrization of a curve is crucial for simplifying complex geometrical structures into more manageable forms. It involves representing a curve using a parameter, often denoted as \( t \), which traces the curve as it varies.In the solution, the curve \( C \) is parametrized by \( \vec{r}(t) = \langle \cos t, 3 \sin t \rangle \) with \( z \)-values defined by \( z = 5 - 2x - y \). This gives us a way to describe the entire 3D curve using t alone.Understanding parametrization allows us to efficiently compute line integrals and surface integrals by breaking down complex paths into simple, single-variable functions.

Surface Integrals

Surface integrals are similar to line integrals but extended over a surface instead of a path. They assess quantities across a surface within a vector field, such as flux, which is often related to the amount flowing across the surface.Stokes’ Theorem cleverly connects line integrals and surface integrals. It states that \( \oint_C \vec{F} \cdot d\vec{r} = \int_S (abla \times \vec{F}) \cdot \vec{n} \, dS \). This means that the circulation of a vector field around a closed curve \( C \) (line integral) equals the surface integral of the curl of the vector field across the surface \( S \) bounded by \( C \). In our case, because the curl of \( \vec{F} \) simplifies to zero, the circulation along \( C \) is zero. Understanding this theorem bridges the gap between the effects on curves and surfaces and is a cornerstone in vector field analysis.